Modeling Cholera-HIV Syndemic with Treatment?

MA Wenrui,Xamxinur Abdurahman

(College of Mathematics and System Sciences,Xinjiang University,Urumqi Xinjiang 830046,China)

0 Introduction

About three decades ago,HIV began to spread worldwide at an alarming rate.HIV is a blood-borne infections and sexually transmitted disease.The HIV attacks the immune system of the infected person that weakens the individuals’s immune systems,which makes the infected person highly vulnerable to infect diseases which ultimately kill the defenseless person.HIV infection itself does not kill the infected person.The advent of the HIV pandemic has led to dramatic increase in the number of cholera cases worldwide.Despite the development of a number of ef f ective treatment over the past half century,cholera remains one of the most destructive bacterial infections.Cholera is an acute gastro-intestinal infection and waterborne disease,which can lead to severe dehydration and death if patients are not treated promptly.And there are reports have been published which explore an increased risk for cholera in HIV infected people[1].Lorenz von Seidlein and Xuan Yi Wang studied that is HIV infection associated with an increased risk for cholera and got that HIV infected individuals had more than twice the risk of cholera infection than HIV uninfected individuals[2].Whereas single disease models have f l ourished for a long time,co-infection models of infections diseases are now coming to limelight.Many researchers studied the dynamics of HIV-TB co-infection.Cristiana J.Silva and Delf i m F.M.Torres[3]proposed a mathematic model to study TB-HIV syndemic and treatment.Navjot Kaur,Mini Ghosh,S.S.Bhatia[4]considered a mathematical model for the dynamics of HIV/TB co-infection by incorporating both screening and treatment of infectives.S.Mushayabasa,J.M.Tchuenche and C.P.Bhunu[5]proposed a model to study the dynamic of HIV/AIDS and gonorrhea co-infection.Inspired by the papers mentioned above and especially by[1]and[3],we provided a in-depth analysis of the qualitative dynamics of cholera-HIV co-infection model.

1 Mathematical model formulation

The total human population is divided into susceptible individuals(S),to both the diseases HIV and cholera,infected with cholera only(IC),the recovered from cholera with permanent immunity(RC),HIV infected not in the AIDS stage of disease progression(IH),HIV infected and in the AIDS stage of disease progression(AH),those dually infected with cholera and HIV not yet displaying AIDS symptoms(ICH),those dually infect with cholera and HIV in the AIDS stage of disease progression(ACH),AIDS patients singly infected with HIV and are on antiretroviral therapy(AHT),AIDS patients dually-infected with HIV and cholera who are on both antiretroviral and cholera treatment(ACHT),those who have recovered from cholera and HIV positive without AIDS symptoms(RCH),those who have recovered from cholera and are infected with HIV in the AIDS stage of disease progression(RCA).Thus,the total human population is given byN=S+IC+RC+IH+ICH+AH+ACH+AHT+ACHI+RCH+RCA.The pathogen of cholera population is given byP.It is assumed that susceptible humans are recruited into the population at a constant rate Λ,and infected with the V.cholera at a rate λc.The infection rate λcis given by,where%is the exposure to contaminated water per unit time,kis the concentration of V.cholerae in water.μis the natural death rate in the human classes.The pathogen population is generated at a rate ηPand its growth is enhanced by the cholera infected individualsICat a constant rate σ,which is a measure of the contribution of in the aquatic environment.The cholera pathogen has a natural death rate μ0in the aquatic environment,which in this case is assume thatμ0> η.Further,susceptible individuals infected with HIV at a rate λH,where

where β is the ef f ective contact rate for HIV transmission,the modif i cation parameterki>1,i=1,2,3.0≤?≤?≤1 due to the fact that treatment reduced the viral load for these individuals.The individuals ofICis generated following infection at a rate λC,and this population decrease following the recovery of infected after treatment at a rate φ1.Individuals in this class acquire HIV infection at a rate αλH,where α >1.The individuals ofIHis generated following infection at a rate λHand by the recovery from cholera infection after treatment by individuals inICHat a rate φ2.Individuals in this class acquire cholera infection at a rate ελc,where ε>1.This population is further decreased following progression to AIDS at a rate γ.The individuals ofAHis generated following the progression to AIDS by individuals inIHat a rate γ,and recovery from cholera infection after treatment ofACHat a rate φ3.Individuals in this class also acquire cholera infection at a rate ελcand die of AIDS-related illness at a rated δ.The individuals inACHis generated by progression to individuals inICHat a ratekγ,wherek>1.AIDS patients are treated at a rate ω,either singly or dually-infected,and dually-infected individuals are assumed to recovery from cholera epidemic at a constant rate φ4.AIDS patients eventually succumb to HIV mortality as the drug wanes out at a reduced rate 0<τ<1.Individuals inICHrecover from the cholera infection at a rate ρ1.Individuals inRCAclass progress into their respective AIDS classRCAat a rate ρ2.We get the nonlinear dif f erential equations(1).The initial conditions of model(1)is as follows:S(0)=S0≥0,IC(0)=IC0>0,RC(0)=RC0>0,P(0)=P0>0,IH(0)=IH0>0,AH(0)=AH0>0,ICH(0)>ICH0>0,ACH(0)=AAH0>0,AHT(0)=AHT0>0,ACHT(0)=ACHT0>0,RCH(0)=RCH0>0,RCA(0)=RCA0>0.

2 HIV-only sub-model

Consider the HIV-only sub-model

2.1 The stability of the disease-free equilibrium for HIV-only sub-model

For system(2)it can be shown that the regionis positively invariant and attracting.The HIV-only model(2)has a disease-free equilibrium point given by.Following[6],it can be shown that the reproduction number for model system(2)is

Therefore,the disease-free equilibrium system(2)is locally asymptotically stable if RH<1,and unstable if RH>1.Lemma 1[7]If system(2)can be written in the form

whereX∈Rmdenotes(its components)the number of infected individuals including latent,infectious,etc.denotes the disease-free equilibrium of the system.

And assume that

(i)For,X?is globally asymptotically stable,

(ii),whereA=DZG(X?,0)is an M-matrix(the of fdiagonal elements ofAare nonnegative)and ΦHis the region where the model makes biological sense.Then the f i xed pointis a globally asymptotic stable equilibrium of model system(2)provided that RH<1.

Applying Lemma 1 to model system(2)gives

Since,SH≤SH+IH+AH+AHT,thus?G(X,Z)≥0,is globally asymptotically stable.We summarise result in Theorem1.

Theorem 1The disease-free equilibrium of the HIV-only model is globally asymptotically stable whenever RH<1,and unstable if RH>1.

2.2 The stability of the endemic equilibrium for HIV-only model

The endemic equilibrium of the HIV-only model is given by,where

Note that,δγ(μ+τδ+τω)?(μ+γ)(μ+τδ)(ω+μ+δ)= ?μ(μ+τδ)(ω+μ+γ+δ)?μωγ <0,so,,therefore,exists if and only if RH>1 and is unique.We now employ the Center Manifold theory[8]to establish the local asymptotic stability of.Let us make the following change of variables in order to apply the Center Manifold theory：SH=x1,IH=x2,AH=x3,AHT=x4,we now use the vector notationX=(x1,x2,x3,x4)T.Then,model system(2)can be written in the form,where

The Jacobian matrix of system(3)atis given by

If β is taken as a bifurcation parameter and consider the case RH=1 and solve for β gives

Note that the linear system of the transformed equation(3)with β = β?,has a simple zero eigenvalue.It can be shown that the Jacobian of(3)at β = β?has a right eigenvector associated with the zero eigenvalue given byu=(u1,u2,u3,u4)T,where

And a left eigenvectorv=(v1,v2,v3,v4),where

Therefore,

Thus,the endemic equilibriumis locally asymptotically stable for RH>1,but close to 1.

Theorem 2The endemic equilibriumis locally asymptotically stable for RH>1,but close to 1.

3 Cholera-only sub-model

The cholera-only sub-model as follows

3.1 The stability of the disease-free equilibrium for cholera-only sub-model

It can be known that:is a positively invariant set.Although there are four states in the model,RCdoes not appear in other three equations,thus we study the system:

The disease free equilibrium of the model system(5)is given by.The basic reproduction number is

Following[6],the disease-free equilibriumis locally asymptotically stable if RC<1 and unstable if RC>1.And we have the following result on the global stability of the disease free equilibrium.

Theorem 3The disease-free equilibriumof(5)is globally asymptotically stable if RC<1.

ProofWe use the comparison theorem to prove the theorem.The rate of the variables representing the infected components of system(5)can be rewritten as

However,since,we havefor allt≥ 0 in ?.Thus,

Given that all the eigenvalues of the matrixF?Vhave negative real parts,it follows that the inequality is stable for RC<1.It thus follows that(IC,P)→(0,0),ast→∞.Thus,sois globally stable.

3.2 The stability of the endemic equilibrium for cholera-only sub-model

The endemic equilibrium is given by,where

?exists if and only if RC>1 and is unique.Using the center manifold theory[8],it is easy to know that

Based on[8],the endemic equilibriumis locally asymptotically stable for RC>1 near 1.

Theorem 4The endemic equilibrium of the cholera-only model is globally asymptotically stable in ?/?0whenever RC>1.

ProofNote,f i rst of all,that the unique endemic equilibrium exists only if RC>1.Further,.Thus,usingand substituting in for the sub model and gives the following limiting system:

Using the Dulac’s multiplier,it follows that

3.3 Sensitivity analysis of RC

Here,the reproductive number RCis analyzed to determine whether or not treatment of cholera patients(modeled by the rate φ1)can lead to the ef f ective control or elimination of cholera in the community.The elasticity of RCwith respect to φ1can be computed as follows:

4 Analysis of the HIV-Cholera model

Having analyzed the dynamics of the two sub-models,the full HIV-cholera model(1)is now considered.First consider the disease free equilibrium which is given by

Using the next generation method[6],it can be shown that the reproduction number for the full HIV-cholera model(1)denoted by RCHis given by

Therefore,the disease-free equilibriumof system(1)is locally asymptotically stable if RCH<1 and unstable if RCH>1.Explicitexpressionsfor the co-infection endemic equilibriumis verycomplicated to compute analytically.Thus,in next section,we will consider a numerical example to show the existence and local stability offor RCH>1.

5 Numerical simulations

In the present section,numerical simulation are carried out to illustrate the analytical results.

When the parameters are chosen as follows: Λ =5,β =0.9,κ2=0.8,? =0.4,μ =0.03,γ =0.4,ω =0.2,δ =0.001,τ=0.8,then we get RH=53.78>1.Fig 1 shows that the the global asymptotically stability of the endemic equilibriumwhen RH>1.Let us choose the parameters as follows Λ =5,%=0.9,β =0.9,κ1=1,κ2=1.2,κ3=1.1,τ=0.2,μ =0.2,? =0.2,η =0.1,σ =1,κ=3,? =0.5,α =1.4,φ1=0.6,μ0=0.8,ε=1,γ =0.3,φ3=0.5,ω =0.5,ρ1=0.3,φ4=0.6,ρ2=0.4,δ=0.4,φ2=0.4.We observe that the state variables converge towhent→∞.In this case,RC=9.375>1,RH=1.4274>1,therefore,RCH=9.375>1.Fig 2 shows that,for RCH>1,the co-infection equilibriumexists and is locally asymptotically stable.

Fig 1 The global asymptotically stability of

Fig 2 The locally asymptotically stable of

References：

[1]Mushayabasa S,Bhunu C P.Is HIV infection associated with an increased risk for cholera?Insights from a mathematical model[J].Biosystems,2012,109:203-213.

[2]Lorenz von Seidlein,Wang Xuanyi,Arminda Macuamule.Is HIV infection associated with an increased risk for cholera?Findings from a case-control study in Mozambique[J].Tropical Medicine and International Health,2008,13(5):683-688.

[3]Navjot Kaur,Mini Ghosh,Bhatia S S.The role of screening and treatment in the transmission dynamics of HIV/AIDS and tuberculosis co-infection:a mathematical study[J].Biology and Physic,2014,40:139-166.

[4]Sunita Gakkhar,Nareshkumar Chavda.A dynamical model for HIV-TB co-infection[J].Applied Mathematics and Computation,2012,218:9261-9270.

[5]Mushayabasa S,Tchuenche J M,Bhunu C P.Modeling gonorrhea and HIV co-interaction[J].BioSystems,2011,103:27-37.[6]Van den Driessche P,James Watmough.Reproduction numbers and sub-threshold endemic equilivria for compartmental models of disease transmission[J].Mathematical Biosciences,2002,180:29-48.

[7]Castillo-Chavez C,Song B.Dynamical models of Tuberculosis and their applications[J].Mathematical Biosciences,2004,1(2):361.

[8]Zindoga Mukandavire,Prasenjit Das.Global analysis of an HIV/AIDS epidemic model[J].World Journal of Modelling and Simulation,2010,3:231-240.